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Theorem splcl 11736
Description: Closure of the substring replacement operator. (Contributed by Stefan O'Rear, 26-Aug-2015.)
Assertion
Ref Expression
splcl  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( S splice  <. F ,  T ,  R >. )  e. Word  A )

Proof of Theorem splcl
Dummy variables  s 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2924 . . . 4  |-  ( S  e. Word  A  ->  S  e.  _V )
2 otex 4388 . . . 4  |-  <. F ,  T ,  R >.  e. 
_V
3 id 20 . . . . . . . 8  |-  ( s  =  S  ->  s  =  S )
4 fveq2 5687 . . . . . . . . . 10  |-  ( b  =  <. F ,  T ,  R >.  ->  ( 1st `  b )  =  ( 1st `  <. F ,  T ,  R >. ) )
54fveq2d 5691 . . . . . . . . 9  |-  ( b  =  <. F ,  T ,  R >.  ->  ( 1st `  ( 1st `  b
) )  =  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) )
65opeq2d 3951 . . . . . . . 8  |-  ( b  =  <. F ,  T ,  R >.  ->  <. 0 ,  ( 1st `  ( 1st `  b ) )
>.  =  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
)
73, 6oveqan12d 6059 . . . . . . 7  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( s substr  <. 0 ,  ( 1st `  ( 1st `  b
) ) >. )  =  ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) )
8 simpr 448 . . . . . . . 8  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  b  =  <. F ,  T ,  R >. )
98fveq2d 5691 . . . . . . 7  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( 2nd `  b
)  =  ( 2nd `  <. F ,  T ,  R >. ) )
107, 9oveq12d 6058 . . . . . 6  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( ( s substr  <. 0 ,  ( 1st `  ( 1st `  b
) ) >. ) concat  ( 2nd `  b ) )  =  ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) ) )
11 simpl 444 . . . . . . 7  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  s  =  S )
128fveq2d 5691 . . . . . . . . 9  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( 1st `  b
)  =  ( 1st `  <. F ,  T ,  R >. ) )
1312fveq2d 5691 . . . . . . . 8  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( 2nd `  ( 1st `  b ) )  =  ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) )
1411fveq2d 5691 . . . . . . . 8  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( # `  s
)  =  ( # `  S ) )
1513, 14opeq12d 3952 . . . . . . 7  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  <. ( 2nd `  ( 1st `  b ) ) ,  ( # `  s
) >.  =  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  (
# `  S ) >. )
1611, 15oveq12d 6058 . . . . . 6  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( s substr  <. ( 2nd `  ( 1st `  b ) ) ,  ( # `  s
) >. )  =  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. ) )
1710, 16oveq12d 6058 . . . . 5  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( ( ( s substr  <. 0 ,  ( 1st `  ( 1st `  b ) ) >.
) concat  ( 2nd `  b
) ) concat  ( s substr  <.
( 2nd `  ( 1st `  b ) ) ,  ( # `  s
) >. ) )  =  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >. ) concat  ( 2nd `  <. F ,  T ,  R >. ) ) concat  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  (
# `  S ) >. ) ) )
18 df-splice 11682 . . . . 5  |- splice  =  ( s  e.  _V , 
b  e.  _V  |->  ( ( ( s substr  <. 0 ,  ( 1st `  ( 1st `  b
) ) >. ) concat  ( 2nd `  b ) ) concat  ( s substr  <. ( 2nd `  ( 1st `  b ) ) ,  ( # `  s
) >. ) ) )
19 ovex 6065 . . . . 5  |-  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) ) concat  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. ) )  e. 
_V
2017, 18, 19ovmpt2a 6163 . . . 4  |-  ( ( S  e.  _V  /\  <. F ,  T ,  R >.  e.  _V )  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) ) concat  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. ) ) )
211, 2, 20sylancl 644 . . 3  |-  ( S  e. Word  A  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) ) concat  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. ) ) )
2221adantr 452 . 2  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) ) concat  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. ) ) )
23 swrdcl 11721 . . . . 5  |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
)  e. Word  A )
2423adantr 452 . . . 4  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
)  e. Word  A )
25 ot3rdg 6322 . . . . . 6  |-  ( R  e. Word  A  ->  ( 2nd `  <. F ,  T ,  R >. )  =  R )
2625adantl 453 . . . . 5  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( 2nd `  <. F ,  T ,  R >. )  =  R )
27 simpr 448 . . . . 5  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  R  e. Word  A )
2826, 27eqeltrd 2478 . . . 4  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( 2nd `  <. F ,  T ,  R >. )  e. Word  A )
29 ccatcl 11698 . . . 4  |-  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
)  e. Word  A  /\  ( 2nd `  <. F ,  T ,  R >. )  e. Word  A )  -> 
( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) )  e. Word  A
)
3024, 28, 29syl2anc 643 . . 3  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) )  e. Word  A
)
31 swrdcl 11721 . . . 4  |-  ( S  e. Word  A  ->  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. )  e. Word  A
)
3231adantr 452 . . 3  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  (
# `  S ) >. )  e. Word  A )
33 ccatcl 11698 . . 3  |-  ( ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) )  e. Word  A  /\  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  (
# `  S ) >. )  e. Word  A )  ->  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) ) concat  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. ) )  e. Word  A )
3430, 32, 33syl2anc 643 . 2  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >. ) concat  ( 2nd `  <. F ,  T ,  R >. ) ) concat  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  (
# `  S ) >. ) )  e. Word  A
)
3522, 34eqeltrd 2478 1  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( S splice  <. F ,  T ,  R >. )  e. Word  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916   <.cop 3777   <.cotp 3778   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   0cc0 8946   #chash 11573  Word cword 11672   concat cconcat 11673   substr csubstr 11675   splice csplice 11676
This theorem is referenced by:  efglem  15303  efgtf  15309  frgpuplem  15359  psgnunilem2  27286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-hash 11574  df-word 11678  df-concat 11679  df-substr 11681  df-splice 11682
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